Spec(Z) and three-manifolds
The following question comes from an old friend, John Baez. John is a renowned mathematical physicist and internet guru. You should look at his website and his blog for an enormously informative and...
View ArticleFundamental groups
Yesterday, at the last algebraic number theory lecture of the term, I defined the fundamental group \pi_1(O_K) of the ring of algebraic integers O_K in an algebraic number field K as the Galois group...
View ArticleNon-commutative constructions
In June of next year, there will be a small workshop at UCL with the title `non-commutative constructions in arithmetic and geometry.’ I will write later in more detail about the goals of the meeting,...
View ArticleFundamental groups and Diophantine geometry
A few weeks ago, I gave a colloquium lecture at Leeds university and subsequently wrote up an exposition based on it. It’s still not entirely `popular,’ but may give a somewhat better sense than my...
View ArticleKazuya Kato
The great arithmetician Kazuya Kato visited twice over the last few weeks, so I thought I’d use the occasion to recommend some writings. An undergraduate level textbook on number theory is Number...
View ArticleKings lecture
Since arriving in the UK, I gave colloquium lectures at QMUL, Leeds, Durham, and Exeter. A `colloquium’ is when you lecture for about an hour to the whole mathematics faculty on the topic of your...
View ArticleSNU Topology of Number Fields
Here are some documents to help you with the course. 1. General Motivation: Lecture at the Cambridge workshop on non-abelian fundamental groups in arithmetic geometry (2009) Lecture at Bordeaux...
View ArticleSNU Topology of number fields, week of 15 July
I was looking over the notes and saw a few items omitted by Milne. First, in the proof of Hilbert’s theorem 90, there is no proof of the linear independence of distinct characters on a group. You can...
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